Transversely isotropic elasticity imaging of cancellous bone

To measure spatial variations in mechanical properties of biological materials, prior studies have typically performed mechanical tests on excised specimens of tissue. Less invasive measurements, however, are preferable in many applications, such as patient-specific modeling, disease diagnosis, and tracking of age- or damage-related degradation of mechanical properties. Elasticity imaging (elastography) is a nondestructive imaging method in which the distribution of elastic properties throughout a specimen can be reconstructed from measured strain or displacement fields. To date, most work in elasticity imaging has concerned incompressible, isotropic materials. This study presents an extension of elasticity imaging to three-dimensional, compressible, transversely isotropic materials. The formulation and solution of an inverse problem for an anisotropic tissue subjected to a combination of quasi-static loads is described, and an optimization and regularization strategy that indirectly obtains the solution to the inverse problem is presented. Several applications of transversely isotropic elasticity imaging to cancellous bone from the human vertebra are then considered. The feasibility of using isotropic elasticity imaging to obtain meaningful reconstructions of the distribution of material properties for vertebral cancellous bone from experiment is established. However, using simulation, it is shown that an isotropic reconstruction is not appropriate for anisotropic materials. It is further shown that the transversely isotropic method identifies a solution that predicts the measured displacements, reveals regions of low stiffness, and recovers all five elastic parameters with approximately 10% error. The recovery of a given elastic parameter is found to require the presence of its corresponding strain (e.g., a deformation that generates ??? is necessary to reconstruct C????), and the application of regularization is shown to improve accuracy. Finally, the effects of noise on reconstruction quality is demonstrated and a signal-to-noise ratio (SNR) of 40 dB is identified as a reasonable threshold for obtaining accurate reconstructions from experimental data. This study demonstrates that given an appropriate set of displacement fields, level of regularization, and signal strength, the transversely isotropic method can recover the relative magnitudes of all five elastic parameters without an independent measurement of stress. The quality of the reconstructions improves with increasing contrast, magnitude of deformation, and asymmetry in the distributions of material properties, indicating that elasticity imaging of cancellous bone could be a useful tool in laboratory studies to monitor the progression of damage and disease in this tissue.     

A theoretical model to predict distribution of the fabric tensor and apparent density in cancellous bone

 The adaptation of cancellous bone to mechanical forces is well recognized. Theoretical models for predicting cancellous bone architecture have been developed and have mainly focused on the distribution of trabecular mass or the apparent density. The purpose of this study was to develop a theoretical model which can simultaneously predict the distribution of trabecular orthotropy/orientation, as represented by the fabric tensor, along with apparent density. Two sets of equations were derived under the assumption that cancellous bone is a biological self-optimizing material which tends to minimize strain energy. The first set of equations provide the relationship between the fabric tensor and stress tensor, and have been verified to be consistent with Wolff’s law of trabecular architecture, that is, the principal directions of the fabric tensor coincide with the principal stress trajectories. The second set of equations yield the apparent density from the stress tensor, which was shown to be identical to those obtained based on local optimization with strain energy density of true bone tissue as the objective function. These two sets of equations, together with elasticity field equations, provide a complete mathematical formulation for the adaptation of cancellous bone. Received: 25 February 1997/Revised version: 23 September 1997     

The contribution of cancellous bone to long bone strength and rigidity

A recent article (Burr and Piotrowski, 1982) suggested that structural analyses of long bone cross-sectional geometry will be inaccurate and should be considered inappropriate when cancellous bone accounts for 10-15% or more of the cross-sectional area. Consideration of material property differences between compact and cancellous bone, however, indicates that even significant proportions of cancellous bone (10-40% of total cross-sectional area) will very likely have negligible effects on bone strength and rigidity, and can be effectively ignored in geometrical analyses of diaphyseal sections. In metaphyseal and epiphyseal regions, however, geometric analyses of section properties such as area moments of inertia are inappropriate, both because of significant trabecular bone effects, and because of the inherent constraints of mechanical beam models.     

Influence of boundary conditions on computed apparent elastic properties of cancellous bone

High-resolution finite element models of trabecular bone can be used to study trabecular structure–function relationships, elasticity, multiaxial strength, and tissue remodelling in more detail than experiments. Beside effects of the model size, scan/analysis resolution, segmentation process, etc., the type of the applied boundary conditions (BCs) have a strong influence on the predicted elastic properties. Appropriate BCs have to be applied on hexahedral digital finite element models in order to obtain effective elastic properties. Homogeneous displacement BCs as proposed by Van Rietbergen et al. (J Biomech 29(12):1653–1657, 1996) lead to “apparent” rather than to “effective” elastic properties. This study provides some answers concerning such differences by comparing various BC types (uniform displacement, mixed BCs, periodic BCs), different volume element definitions (original and mirrored models), and several bone volume fractions (BVTV ranging from 6.5 to 37.6%). First, the mixed BCs formulated by Hazanov (Arch Appl Mech 68(6):385–394, 1998) are theoretically extended to shear loading of a porous media. Second, six human bone samples are analyzed, their orthotropic Young’s moduli, shear moduli, and Poisson’s ratios computed and compared. It is found that the proposed mixed BCs give exactly the same effective elastic properties as periodic BCs if a periodic and orthotropic micro-structured material is used and thus denoted as “periodicity compatible” mixed uniform BCs (PMUBCs). As bone samples were shown to be nearly orthotropic for volume element side lengths ≥5 mm the proposed mixed BCs turn out to be the best choice because they give again essentially the same overall elastic properties as periodic BCs. For bone samples of smaller dimensions ( < 5 mm) with a strong anisotropy (beyond orthotropy) uniform displacement BCs remain applicable but they can significantly overestimate the effective stiffness. In Memoriam, Prof. Christian Huet.     

Finite element calculated uniaxial apparent stiffness is a consistent predictor of uniaxial apparent strength in human vertebral cancellous bone tested with different boundary conditions

Strong correspondence between the uniaxial apparent strength and stiffness of cancellous bone allows the use of stiffness as a predictor of bone strength. Measured values of mechanical properties in cancellous bone can be different between experiments due to different experimental conditions. In the current study, bone volume fraction, experimentally determined and finite element (FE) predicted stiffness were examined as predictors of cancellous bone ultimate strength in two different groups each of which was tested using a different end constraint. It is demonstrated that, although always significant, the relationships of strength with bone volume fraction and experimentally determined stiffness are different between test groups. Apparent stiffness, estimated by FE modeling, predicts the ultimate strength of human cancellous bone consistently for all examined experimental protocols.     

The fabric dependence of the orthotropic elastic constants of cancellous bone

It has been proposed that the orthotropic elastic constants of cancellous bone depend upon a tensorial measure of anisotropy called fabric as well as the tissue's structural density. Cowin (1985, Mechanics Mater, 4, 137-147; 1986, J. biomech. Engng 108, 83-88) developed explicit relationships for the elastic constant, structural density and fabric relationship. In this study the orthotropic elastic moduli, structural density, and fabric components were measured for 11 cancellous bone specimens from five bovine femora and for 75 specimens from three human proximal tibiae and fitted to these relationships using a least squares analysis. The relationships explained between 72 and 94% of the variance in the elastic constants. The relationships between the elastic constants and squared or cubed power functions of structural density had better predictive value over the entire distribution of the data than did expressions with linear functions of structural density.     

On the meaning of density dependence

Summary Despite a long history, the term density dependence lacks a generally accepted definition. A definition is offered that seems consistent with most other definitions and general usage, that is, a density-dependent factor is any component of the environment whose intensity is correlated with population density and whose action affects survival and reproduction. This definition is used in evaluating the role of territorial behavior, the availability of nest sites, and competition in determining the size of a population. Because neither territory size nor the number of nest sites is correlated with either density or with changes in the birth and death rates of these populations, these cannot be considered density-dependent factors. Competition determines who does breed and who does not rather than the number of breeders, and thus it is not a density-dependent factor determining a population's size.     

Tensile properties of rat femoral bone as functions of bone volume fraction, apparent density and volumetric bone mineral density

Mechanical testing has been regarded as the gold standard to investigate the effects of pathologies on the structure-function properties of the skeleton. Tensile properties of cancellous and cortical bone have been reported previously; however, no relationships describing these properties for rat bone as a function of volumetric bone mineral density (ρ(MIN)), apparent density or bone volume fraction (BV/TV) have been reported in the literature. We have shown that at macro level, compression and torsion properties of rat cortical and cancellous bone can be well described as a function of BV/TV, apparent density or ρ(MIN) using non-destructive micro-computed tomographic imaging and mechanical testing to failure. Therefore, the aim of this study is to derive a relationship expressing the tensile properties of rat cortical bone as a function of BV/TV, apparent density or ρ(MIN) over a range of normal and pathologic bones. We used bones from normal, ovariectomized and osteomalacic animals. All specimens underwent micro-computed tomographic imaging to assess bone morphometric and densitometric indices and uniaxial tension to failure. We obtained univariate relationships describing 74-77% of the tensile properties of rat cortical bone as a function of BV/TV, apparent density or ρ(MIN) over a range of density and common skeletal pathologies. The relationships reported in this study can be used in the structural rigidity to provide a non-invasive method to assess the tensile behavior of bones affected by pathology and/or treatment options.     

Femoral head apparent density distribution predicted from bone stresses

A new theory relating bone morphology to applied stress is used to predict the apparent density distribution in the femoral head and neck. Cancellous bone is modeled as a self-optimizing material and cortical bone as a saturated (maximum possible bone density) response to stress in the bone tissue. Three different approaches are implemented relating bone apparent density to: (1) the von Mises stress, (2) the strain energy density in the mineralized tissue and (3) a defined closed effective stress (spherical stress). An iterative nonlinear three-dimensional finite element model is used to predict the apparent density distribution in the femoral head and neck for each of the three approaches. It is shown that the von Mises stress (an open effective stress) cannot accurately predict bone apparent density. It is shown that strain energy density and the defined closed effective stress can predict apparent density and that they give predictions consistent with the observed density pattern in the femoral head and neck.     

The dependence of transversely isotropic elasticity of human femoral cortical bone on porosity

The objective of this study was to examine the dependence of the elastic properties of cortical bone as a transversely isotropic material on its porosity. The longitudinal Young's modulus, transverse Young's modulus, longitudinal shear modulus, transverse shear modulus, and longitudinal Poisson's ratio of cortical bone were determined from eighteen groups of longitudinal and transverse specimens using tensile and torsional tests on a servo-hydraulic material testing system. These cylindrical waisted specimens of cortical bone were harvested from the middle diaphysis of three pairs of human femora. The porosity of these specimens was assessed by means of histology. Our study demonstrated that the longitudinal Young's and shear moduli of human femoral cortical bone were significantly (p<0.01) negatively correlated with the porosity of cortical bone. Conversely, the elastic properties in the transverse direction did not have statistically significant correlations with the porosity of cortical bone. As a result, the transverse elastic properties of cortical bone were less sensitive to changes in porosity than those in the longitudinal direction. Additionally, the anisotropic ratios of cortical bone elasticity were found to be significantly (p<0.01) negatively correlated with its porosity, indicating that cortical bone tended to become more isotropic when its porosity increased. These results may help a number of researchers develop more accurate micromechanics models of cortical bone.     

Anisotropy and inhomogeneity of the trabecular structure can describe the mechanical strength of osteoarthritic cancellous bone

Osteoarthritic cancellous bone was studied to investigate the development of this pathology, and the functional changes it induces in the bone. In order to predict how the morphological alterations of the tissue induced by the pathology can change the mechanical properties of the structure, two different strategies have been used in the literature: (1) emphasising the influence of structural anisotropy; (2) stressing the highly inhomogeneous characteristics of cancellous bone. The aim of the present study was to verify the theory that mechanical strength of osteoarthritic cancellous bone depends both on tissue anisotropy and inhomogeneity.Twenty-five specimens were extracted from osteoarthritic femoral heads, along selected directions, and analysed by means of a microtomograph. The same specimens were mechanically tested in compression to determine the mechanical strength. The most representative structural parameters, confirmed by a stepwise analysis, were used to define four models to describe the measured mechanical strength. The models were applied neglecting (global analysis) or considering (local analysis) tissue inhomogeneities to verify whether the correlation with ultimate stress could be improved.The coefficient of determination increased from 0.53, considering only bone volume fraction, up to 0.88, combining it with off-axis angle and normalised eigenvalue. A further improvement was found performing a local analysis (R²=0.90), which corresponded to a decrease of 17% in the residual error.The proposed approach of considering both tissue anisotropy and inhomogeneity improved the accuracy in predicting the mechanical behaviour of cancellous bone tissue and should be suitable for more general loading conditions.     

Effect of alendronate administration on bone mineral density and bone strength in castrated rats.

Castration of male rats leads to increased bone turnover and osteopenia. This study was conducted to examine the effects of the aminobisphosphonate alendronate on castration-induced bone changes. Bisphosphonates are drugs that inhibit bone turnover by decreasing the resorption. Since they suppress bone remodeling, they may also prevent the repair of microdamage and decrease bone strength. Although the mechanical properties of bones are directly related to the determination of fracture risk, bisphosphonate effects on the related variables have scarcely been investigated. Twenty-four male Wistar rats at two months of age were castrated or sham-operated to evaluate the effects of long-term administration (six months) of sodium alendronate at a dose of 1 mg/kg/day. The bones were tested mechanically by a three-point bending test in a Mini Bionix (MTS) testing system. High bone remodeling seen in castrated rats expressed by increased TrACP and B-ALP was suppressed by alendronate administration. Bone from castrated rats was characterized by a reduction in bone density as well as ash, calcium and phosphate content. Castration significantly altered mechanical properties of bone and femoral cortical thickness. When castrated rats were treated with high dose of alendronate, the changes in bone density resulting from castration were entirely prevented, and mechanical analysis revealed preserved mechanical strength of femur and cortical thickness. We conclude that castration induces cortical bone loss associated with high bone turnover in the male rat, and this bone loss can be prevented by alendronate through the inhibition of osteoclastic activity, while preserving the mechanical properties of bone. These results document the efficacy of alendronate, even at high doses, in preventing bone loss, loss of bone mechanical strength, and the rise in biochemical bone turnover indicators due to castration in rats, and raises the possibility that a alendronate could be equally effective in humans.     

Compressive axial mechanical properties of rat bone as functions of bone volume fraction,apparent density and micro-ct based mineral density

Mechanical testing has been regarded as the gold standard to investigate the effects of pathologies on the structure–function properties of the skeleton. With recent advances in computing power of personal computers, virtual alternatives to mechanical testing are gaining acceptance and use. We have previously introduced such a technique called structural rigidity analysis to assess mechanical strength of skeletal tissue with defects. The application of this technique is predicated upon the use of relationships defining the strength of bone as a function of its density for a given loading mode. We are to apply this technique in rat models to assess their compressive skeletal response subjected to a host of biological and pharmaceutical stimulations. Therefore, the aim of this study is to derive a relationship expressing axial compressive mechanical properties of rat cortical and cancellous bone as a function of equivalent bone mineral density, bone volume fraction or apparent density over a range of normal and pathologic bones.We used bones from normal, ovariectomized and partially nephrectomized animals. All specimens underwent micro-computed tomographic imaging to assess bone morphometric and densitometric indices and uniaxial compression to failure.We obtained univariate relationships describing 71–78% of the mechanical properties of rat cortical and cancellous bone based on equivalent mineral density, bone volume fraction or apparent density over a wide range of density and common skeletal pathologies. The relationships reported in this study can be used in the structural rigidity analysis introduced by the authors to provide a non-invasive method to assess the compressive strength of bones affected by pathology and/or treatment options.     

Variability of tissue mineral density can determine physiological creep of human vertebral cancellous bone

Creep is a time-dependent viscoelastic deformation observed under a constant prolonged load. It has been indicated that progressive vertebral deformation due to creep may increase the risk of vertebral fracture in the long-term. The objective of this study was to examine the relationships of creep with trabecular architecture and tissue mineral density (TMD) parameters in human vertebral cancellous bone at a physiological static strain level. Architecture and TMD parameters of cancellous bone were analyzed using microcomputerized tomography (micro-CT) in specimens cored out of human vertebrae. Then, creep and residual strains of the specimens were measured after a two-hour physiological compressive constant static loading and unloading cycle. Creep developed (3877 ± 2158 με) resulting in substantial levels of non-recoverable post-creep residual strain (1797 ± 1391 με). A strong positive linear correlation was found between creep and residual strain (r = 0.94, p < 0.001). The current results showed that smaller thickness, larger surface area, greater connectivity of trabeculae, less mean tissue mineral density (TMD, represented by gray levels) and higher variability of TMD are associated with increasing logarithmic creep rate. The TMD variability (GL(COV)) was the strongest correlate of creep rate (r = 0.79, p < 0.001). This result suggests that TMD variability may be a useful parameter for estimating the long-term deformation of a whole vertebral body. The results further suggest that the changes in TMD variability resulting from bone remodeling are of importance and may provide an insight into the understanding of the mechanisms underlying progressive failure of vertebral bodies and development of a clinical fracture.     

Estimation of bone matrix apparent stiffness variation caused by osteocyte lacunar size and density

The role of osteocyte lacunar size and density on the apparent stiffness of bone matrix was predicted using a mechanical model from the literature. Lacunar size and lacunar density for different bones from different gender and age groups were used to predict the range of matrix apparent stiffness values for human cortical and cancellous tissue. The results suggest that bone matrix apparent stiffness depends on tissue type (cortical versus cancellous), age, and gender, the magnitudes of the effects being significant but small in all cases. Males had a higher predicted matrix apparent stiffness than females for vertebral cancellous bone (p< I0(-7)) and the difference increased with age (p =0.0007). In contrast, matrix apparent stiffness was not different between males and females forfemoral cortical bone and increased with age in both males (p < 0.0001) and females (p < 0.0364). Osteocyte lacunar density and size may cause significant gender and age-related variations in bone matrix apparent stiffness. The magnitude of variations in matrix apparent stiffness was small within the physiological range of lacunar size and density for healthy bone, whereas the variations can be profound in certain pathological cases. It was proposed that the mechanical effects of osteocyte density be uncoupled from their biological effects by controlling lacunar size in normal bone.     

Tensile and compressive properties of cancellous bone

The relationship between the mechanical properties of trabecular bone in tension and compression was investigated by non-destructive testing of the same specimens in tension and compression, followed by random allocation to a destructive test in either tension or compression. There was no difference between Young's modulus in tension and compression, and there was a strong positive correlation between the values (R = 0.97). Strength, ultimate strain and work to failure was significantly higher in tensile testing than in compressive testing.     

Anisotropy of the fatigue behaviour of cancellous bone

The fatigue behaviour of materials is of particular interest for the failure prediction of materials and structures exposed to cyclic loading. For trabecular bone structures only a few sets of lifetime data have been reported in the literature and structural measures are commonly not considered. The influence of load contributions which are not aligned with the main physiological axis remains unclear. Furthermore site and species dependent relationships are not well described. In this study five different groups of trabecular bone, defined in terms of orientation, species and site were exposed to cyclic compression. In total, 108 fatigue tests were analysed. The lifetimes were found to decrease drastically when off-axis loads were applied. Additionally, species and site strongly affect fatigue lifetimes. Strains at failure were also found to be a function of orientation.